Abstract
A transfer-matrix method (TMM) is presented for the development of concentration and flux profiles in multicomponent diffusion involving any numbern of components. From interdiffusion fluxes or concentration gradients available at an initial positionx s, the authors derive expressions for the transfer matrix and its integral so that the concentrations or interdiffusion fluxes of the components can be obtained at any coordinatex. The TMM requires data for interdiffusion coefficients, which are obtained as average values over selected regions by the method of moments developed by Dayananda. Expressions for the concentrations are also obtained from initial conditions on the fluxes or the concentration gradients. The method is also applicable to the case when all the concentrations are known at two ends of a region over which the diffusion coefficients are considered constant. The integration of the fluxes over time, or over the coordinatex, can be evaluated using the transfer-matrix approach, provided the value of the interdiffusion flux is given at a given coordinate. The TMM is applicable to any number of components and can be regarded as a compact generalization of the solutions available for ternary diffusion couples with constant interdiffusion coefficients. An application of the method is illustrated with the experimental data for a ternary Cu-Ni-Zn diffusion couple, and the results are compared with those based on the Fujita-Gosting solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Onsager, Theories and Problems of Liquid Diffusion,Ann. NY Acad. Sci., 1945,46, p 241–265
A. Fick, Uber Diffusion,Ann. Phys., 1855,170, p 59–86
J.S. Kirkaldy and D.J. Young,Diffusion in the Condensed State, The Institute of Metals, London, 1987, p 226–272
M.A. Dayananda and Y.H. Sohn, A New Analysis for the Determination of Ternary Interdiffusion Coefficients from a Single Diffusion Couple,Metall. Mater. Trans., 1999,30A, p 535–543
M.A. Dayananda, An Analysis of Concentration Profiles for Fluxes, Diffusion Depths, and Zero-Flux Planes in Multicomponent Diffusion,Metall. Trans., 1983,14A, p 1851–1858
M.A. Dayananda, Analysis of Multicomponent Diffusion Couples for Interdiffusion Fluxes and Interdiffusion Coefficients,J. Phase Equilib. Diffus., 2005,26, p 441–446
K.M. Day, L.R. Ram-Mohan, and M.A. Dayananda, Determination and Assessment of Ternary Interdiffusion Coefficients from Individual Diffusion Couples,Journal Phase Equilib. Diffus., 2005,26, p 579–590
L.R. Ram-Mohan and M.A. Dayananda, “MultiDiFlux,” software, https://engineering.purdue.edu/MSE/Fac_Staff/Faculty/dayananda.wshtml, Purdue University, 2005
L.R. Ram-Mohan,Finite Element and Boundary Element Applications in Quantum Mechanics, Oxford University Press, Oxford, UK, 2002
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery,Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1992
L.R. Ram-Mohan, and M.A. Dayananda, “Analysis of Multi-Component Multiphase Metallic Diffusion Couples: Data Fitting and Extraction of Averaged Diffusion Coefficients,” Under preparation for publication, 2006
M.A. Dayananda and C.W. Kim, Zero-Flux Planes and Flux Reversals in Cu-Ni-Zn Diffusion Couples,Metall. Trans., 1979,10A, p 1333–1339
C. Matano, On the Relation between the Diffusion Coefficients and Concentrations of Solid Metals (The Nickel-Copper System),Jpn. J. Phys. (Trans.), 1933,8, p 109–113
L. Boltzmann, Zur Integration der Diffusiongleichung bei Variablen Diffusions-coefficienten,Ann. Phys., 1894,53, p 959–964
L.R. Ram-Mohan, and M.A. Dayananda, A Transfer Matrix Method for the Calculation of Concentrations and Fluxes in Multicomponent Diffusion Couples,Acta Mater., 2006,54, p 2325–2334
C.W. Kim and M.A. Dayananda, Zero-Flux Planes and Flux Reversals in the Cu-Ni-Zn System at 775 °C,Metall. Trans. A, 1984,15A(4), p 649–659
L.A. Pipes and L.R. Harvill,Applied Mathematics for Engineers and Physicists, McGraw-Hill, New York, 1985
L.R. Ram-Mohan, K.H. Yoo, and R.L. Aggarwal, A Transfer Matrix Algorithm for the Calculation of Band Structure of Semiconductor Superlattices,Phys. Rev. B, 1988,38, p 6151–6159
B. Chen, M. Lazzouni, and L.R. Ram-Mohan, The Diagonal Representation for the Transfer Matrix Method for Obtaining Electronic Energy Levels in Layered Semiconductor Heterostructures,Phys. Rev. B, 1992,45, p 1204–1212
R. Taylor and R. Krishna,Multicomponent Mass Transfer, Wiley, New York, 1993
H. Fujita and L.J. Gosting, An Exact Solution of the Equations of Free Diffusion in Three-Component Systems with Interacting Flows, and Its Use in the Evaluation of the Diffusion Coefficients,J. Am. Chem. Soc., 1956,78, p 1099
M.A. Dayananda, Average Effective Interdiffusion Coefficients and the Matano Plane Composition,Metall. Mater. Trans., 1996,27A, p 2504–2509
J.E. Morral, Rate Constants for Interdiffusion,Scr. Met., 1984,18, p 1251–1256
M.S. Thompson and J.E. Morral, The Square Root Diffusivity,Acta Metall., 1986,34, p 2201–2203
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ram-Mohan, L.R., Dayananda, M.A. A transfer-matrix method for analysis of multicomponent diffusion with any number of components. JPED 27, 566–571 (2006). https://doi.org/10.1007/BF02736557
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02736557